Whether or not the President was born in America has never been a matter of genuine controversy among reasonable people because the facts and documents do not cast doubt on the issue. That hasn't stopped "birther" conspiracies from popping up, but none of them are any better supported by the evidence than "moon landing" conspiracies and are a tool of the irrational or the obtuse.
The State of Hawaii releases to citizens a copy of their certificate of live birth which has been called a short form document. If you are a citizen of Hawaii and request a copy of your birth certificate, that is what they give you and that is all that is required by law to prove citizenship in the state, get a passport, run for office, etc. That is what the President provided way back in 2008 during his campaign for office. In fact, the Obama campaign allowed reporters to examine his copy. Here are some pictures of it. Also, reporters researching this found that two newspapers in Honolulu ran birth announcements in August of 1961.
Not satisfied with any of that, birthers demanded to see the original document on file in Hawaii. The state of Hawaii does not release that document, but the director of Hawaii's Department of Health verified that it existed, that she had seen it, and that there was no doubt that then Senator Obama was born in Hawaii. Yet the conspiracy theory that the President was not a citizen continued, no matter how absurd it seemed.
More recently, the President asked for a special exemption of Hawaiian law so that a copy of the long form certificate of live birth that is kept by the Hawaiian Department of Health could be released. This was very unusual and would not have been done if not for the lunacy of birtherism in America. Which I am sad to say is still not gone, even in the face of overwhelming evidence that the President is indeed a natural born US citizen.
I've embedded a lot of links to supporting information in this post. Please follow those links and read those documents. In some cases those web pages have embedded links of their own, so read those too. When I refer to the evidence of the President's citizenship being overwhelming, I'm not indulging in hyperbole.